University of Bath

Quantum limits are objects describing the limit of quadratic quantities (Af_n,f_n) where (f_n) is a sequence of unit vectors in a Hilbert space and A ranges over an algebra of bounded operators. We will discuss the motivation underlying this notion with some important examples from quantum mechanics and from analysis.

Nonlocal minimal surfaces are hypersurfaces of Euclidean space that minimize the fractional perimeter, a geometric functional introduced in 2010 by Caffarelli, Roquejoffre, and Savin in connection with phase transition problems displaying long-range interactions.

In this talk, I will introduce these objects, describe the most important progresses made so far in their analysis, and discuss the most challenging open questions.

I will then focus on the particular case of nonlocal minimal graphs and present some recent results obtained on their regularity and classification in collaboration with X. Cabre, A. Farina, and L. Lombardini.

I will describe new solutions to the stationary Ginzburg-Landau equation in 3 dimensions with vortex lines given by interacting helices, with degree one around each filament and total degree an arbitrary positive integer. I will also present results on the asymptotic behavior of vortices in the entire plane for a dissipative Ginzburg-Landau equation. This is work in collaboration with Manuel del Pino, Remy Rodiac, Maria Medina, Monica Musso and Juncheng Wei.

Description:In his seminal work from the 1950s, William Feller classified all one-dimensional diffusions in terms of their ability to access the boundary (Feller's test for explosions) and to enter the interior from the boundary. Feller's technique is restricted to diffusion processes as the corresponding differential generators allow explicit computations and the use of Hille-Yosida theory. In the present article we study exit and entrance from infinity for jump diffusions driven by a stable process.Many results have been proved for jump diffusions, employing a variety of techniques developed after Feller's work but exit and entrance from infinite boundaries has long remained open. We show that the these processes have features not observes in the diffusion setting. We derive necessary and sufficient conditions on σ so that (i) non-exploding solutions exist and (ii) the corresponding transition semigroup extends to an entrance point at `infinity'. Our proofs are based on very recent developments for path transformations of stable processes via the Lamperti-Kiu representation and new Wiener-Hopf factorisations for Lévy processes that lie therein. The arguments draw together original and intricate applications of results using the Riesz-Bogdan--Żak transformation, entrance laws for self-similar Markov processes, perpetual integrals of Lévy processes and fluctuation theory, which have not been used before in the SDE setting, thereby allowing us to employ classical theory such as Hunt-Nagasawa duality and Getoor's characterisation of transience and recurrence.

For a finite subgroup $G$ of $SL(2,C)$ and for $n \geq 1$,  the Hilbert scheme $X=Hilb^{[n]}(S)$ of $n$ points on the minimal resolution $S$ of the Kleinian singularity $C^2/G$ provides a crepant resolution of the symplectic quotient $C^{2n}/G_n$, where $G_n$ is the wreath product of $G$ with $S_n$. I'll explain why every projective, crepant resolution of $C^{2n}/G_n$ is a quiver variety, and why the movable cone of $X$ can be described in terms of an extended Catalan hyperplane arrangement of the root system associated to $G$ by John McKay. These results extend the algebro-geometric aspects of Kronheimer's hyperkahler description of $S$ to higher dimensions. This is joint work with Gwyn Bellamy.

In a joint-work with Andrea Collevecchio and Vladas Sidoravicius,  we study  phase transitions in the recurrence/transience of a class of self-interacting random walks on trees, which includes the once-reinforced random walk. For this purpose, we define the branching-ruin number of a tree, which is  a natural way to measure trees with polynomial growth and therefore provides a polynomial version of the branching number defined by Furstenberg (1970) and studied by R. Lyons (1990). We prove that the branching-ruin number of a tree is equal to the critical parameter for the recurrence/transience of the once-reinforced random walk on this tree. We will also mention two other results where the branching-ruin number arises as critical parameter: first, in the context of random walks on heavy-tailed random conductances on trees and, second, in the case of Volkov's M-digging random walk.

We prove an a priori bound for solutions of the dynamic $\Phi^4_3$ equation.

This bound provides a control on solutions on a compact space-time set only in terms of the realisation of the noise on an enlargement of this set, and it does not depend on any choice of space-time boundary conditions.

We treat the  large and small scale behaviour of solutions with completely different arguments.For small scales we use bounds akin to those presented in Hairer's theory of regularity structures. We stress immediately that our proof is fully self-contained, but we give a detailed explanation of how our arguments relate to Hairer's. For large scales we use a PDE argument based on the maximum principle. Both regimes are connected by a solution-dependent regularisation procedure.

The fact that our bounds do not depend on space-time boundary conditions makes them useful for the analysis of large scale properties of solutions. They can for example be used in a compactness argument to construct solutions on the full space and their invariant measures

Suppose that we want to minimise the L-infinity norm of the Laplacian of a function (or a similar quantity) under Dirichlet boundary conditions. This is a convex, but not strictly convex variational problem. Nevertheless, it turns out that it has a unique solution, which is characterised by a system of PDEs. The behaviour is thus quite different from the better-known first order problems going back to Aronsson. This is joint work with N. Katzourakis (Reading).
 

We consider the  symbiotic branching model, which describes a spatial population consisting of two types in terms of a coupled system of stochastic PDEs. One particularly important special case is Kimura's stepping stone model in evolutionary biology. Our main focus is a description of the interfaces between the types in the large scale limit of the system. As a new tool we will introduce a moment duality, which also holds for the limiting model. This also has implications for a classification of entrance laws of annihilating Brownian motions.